Kohn-Sham Equations

Using what we have just learned about taking derivatives of real functions of complex variables and including the normality constraint of each orbital with a separate Lagrange multiplier , the condition for the constrained minimization of (12) is

We now take this derivative term by term.

The kinetic energy (10) has only one
term
in it, so the derivative is just what this term multiplies,

For the electron-nuclear potential energy (3), the only
term which depends on
is the charge density, where
multiplies
. The nuclear potential
is unchanged as
varies, so the final term is just

The electron-electron energy has a very similar structure. The only
difference is that, here, when we change
, the
potential function also changes because it depends on
. The net effect of the change in is the same as
that of the direct change in . To see this we note that Poisson's
equation
implies also that
. Thus,

where we have moved the from acting on to acting on by integrating by parts twice. Since both terms are equal, we can take just twice the term,

For the exchange-correlation term, we have already derived in
(14) that
. By the chain rule we just need to multiply this by
for the
result

Fortunately, depends only on the nuclear positions and
does not change with
, so

And, finally, only appears once in the constraint term, making the derivative,

Summing all of these contributions, setting the resulting equation to
zero, moving the ``
'' term to the
right-hand side, and dividing through by , we get the final result,

Fortunately for us, this is in the form of a very well-known equation for which there are standard techniques. This is in the form of the standard Schrödinger equation,

where we define the potential term as

and we define . We interpret the potential as just the sum of the nuclear potential, the electrostatic potential created by the electrons, and an extra, ``exchange-correlation'' potential correction, . Since the Lagrange-multipliers are unknown constants at the start, we may as well think in terms of the instead, which have the interpretation of the Schrödinger energies for each orbital.

Tomas Arias 2004-01-26